Sequences and Series
A sequence is an ordered list of numbers following a pattern. A series is the sum of a sequence. Understanding sequences and series is essential for calculus, where limits of infinite series appear in Taylor series, integrals, and fundamental theorems.
In this lesson
1 What Sequences Are
A sequence is a list of numbers in a specific order, typically defined by a rule. The numbers are called terms: a₁ (first term), a₂ (second term), and so on. The general term aₙ gives the nth term as a formula.
Examples: 2, 4, 6, 8, 10... (even numbers). 1, 1, 2, 3, 5, 8, 13... (Fibonacci). 1, 1/2, 1/4, 1/8... (halving each time). Each has a pattern that can be expressed as a formula for aₙ.
2 Arithmetic Sequences
An arithmetic sequence adds the same number every step. 3, 7, 11, 15 adds 4 each time. The common difference is 4. General term: aₙ = a₁ + (n−1)d
79Sum of an arithmetic sequence (arithmetic series): Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d)
8203 Geometric Sequences
A geometric sequence multiplies by the same number every step. 2, 6, 18, 54 multiplies by 3 each time. The common ratio is 3. General term: aₙ = a₁ × rⁿ⁻¹
4,374Sum of finite geometric series: Sₙ = a₁(1−rⁿ)/(1−r) for r ≠ 1.
4 Series and Summation Notation
Summation notation (Σ) compactly represents sums. Σᵢ₌₁ⁿ aᵢ means "sum the terms aᵢ for i from 1 to n."
Σᵢ₌₁⁵ i² = 1² + 2² + 3² + 4² + 5² = 1+4+9+16+25 = 55
Taylor series express functions as infinite polynomial sums: eˣ = 1 + x + x²/2! + x³/3! + ... This allows functions to be approximated to arbitrary precision, which is fundamental to numerical computation and advanced calculus.
5 Infinite Geometric Series
Add up infinitely many terms and you might expect to get infinity. Sometimes you do. But if each term is smaller than the previous by a consistent ratio less than 1, the total converges to a finite number. S∞ = a₁/(1−r)
21 + 2 + 4 + 8 + ... has r=2. Since |r| > 1, the series diverges , the sum grows without bound. The formula S∞ = a₁/(1−r) only applies when |r| < 1.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.